function [x,w] = legendreNodesWeights(N,a,b)
% This script is for computing definite integrals using Legendre-Gauss 
% Quadrature. Computes the Legendre-Gauss nodes and weights  on an interval
% [a,b] with truncation order N
%
% Suppose you have a continuous function f(x) which is defined on [a,b]
% which you can evaluate at any x in [a,b]. Simply evaluate it at all of
% the values contained in the x vector to obtain a vector f. Then compute
% the definite integral using sum(f.*w);
%
% Written by Greg von Winckel - 02/25/2004


N = N-1;
N1 = N+1; 
N2 = N+2;
xu = linspace(-1,1,N1)';
% Initial guess
y = cos((2*(0:N)'+1)*pi/(2*N+2))+(0.27/N1)*sin(pi*xu*N/N2);
% Legendre-Gauss Vandermonde Matrix
L = zeros(N1,N2);
% Derivative of LGVM
Lp = zeros(N1,N2);
% Compute the zeros of the N+1 Legendre Polynomial
% using the recursion relation and the Newton-Raphson method
y0 = 2;
% Iterate until new points are uniformly within epsilon of old points
while max(abs(y-y0))>eps
    
    
    L(:,1) = 1;
    Lp(:,1) = 0;
    
    L(:,2) = y;
    Lp(:,2) = 1;
    
    for k = 2:N1
        L(:,k+1) = ( (2*k-1)*y.*L(:,k)-(k-1)*L(:,k-1) )/k;
    end
 
    Lp = (N2)*( L(:,N1)-y.*L(:,N2) )./(1-y.^2);   
    
    y0 = y;
    y = y0-L(:,N2)./Lp;
    
end
% Linear map from[-1,1] to [a,b]
x = (a*(1-y)+b*(1+y))/2;      
% Compute the weights
w = (b-a)./((1-y.^2).*Lp.^2)*(N2/N1)^2;